Particle Swarm Optimization as an Efficient Computational Method in order to Minimize Vibrations of Multimesh Gears Transmission

(1)

HAL Id: hal-03257198

https://hal.archives-ouvertes.fr/hal-03257198

Submitted on 10 Jun 2021

HAL is a multi-disciplinary open access

archive for the deposit and dissemination of

sci-entific research documents, whether they are

pub-lished or not. The documents may come from

teaching and research institutions in France or

abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est

destinée au dépôt et à la diffusion de documents

scientifiques de niveau recherche, publiés ou non,

émanant des établissements d’enseignement et de

recherche français ou étrangers, des laboratoires

publics ou privés.

Distributed under a Creative Commons Attribution| 4.0 International License

Particle Swarm Optimization as an Eﬀicient

Computational Method in order to Minimize Vibrations

of Multimesh Gears Transmission

Alexandre Carbonelli, Joël Perret-Liaudet, Emmanuel Rigaud, Alain Le Bot

To cite this version:

Alexandre Carbonelli, Joël Perret-Liaudet, Emmanuel Rigaud, Alain Le Bot. Particle Swarm

Op-timization as an Eﬀicient Computational Method in order to Minimize Vibrations of Multimesh

Gears Transmission. Advances in Acoustics and Vibration, Hindawi Publishing Corporation, 2011,

pp.195642. �10.1155/2011/195642�. �hal-03257198�

(2)

Volume 2011, Article ID 195642,6pages doi:10.1155/2011/195642

Research Article

Particle Swarm Optimization as an Efficient

Computational Method in order to Minimize Vibrations

of Multimesh Gears Transmission

Alexandre Carbonelli, Jo¨el Perret-Liaudet, Emmanuel Rigaud, and Alain Le Bot

Laboratoire de Tribologie et Dynamique des Syst`emes, UMR CNRS 5513, Ecole Centrale de Lyon, Universit`e de Lyon, 36 Avenue Guy de Collongue, 69134 Ecully Cedex, France

Correspondence should be addressed to Alexandre Carbonelli,alexandre.carbonelli@ec-lyon.fr

Received 12 January 2011; Accepted 13 April 2011 Academic Editor: Snehashish Chakraverty

Copyright © 2011 Alexandre Carbonelli et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The aim of this work is to present the great performance of the numerical algorithm of Particle Swarm Optimization applied to find the best teeth modifications for multimesh helical gears, which are crucial for the static transmission error (STE). Indeed, STE fluctuation is the main source of vibrations and noise radiated by the geared transmission system. The microgeometrical parameters studied for each toothed wheel are the crowning, tip reliefs and start diameters for these reliefs. Minimization of added up STE amplitudes on the idler gear of a three-gear cascade is then performed using the Particle Swarm Optimization. Finally, robustness of the solutions towards manufacturing errors and applied torque is analyzed by the Particle Swarm algorithm to access to the deterioration capacity of the tested solution.

1. Introduction

The STE under load [1] is defined as the diﬀerence between

the actual position of the driven gear and its theoretical posi-tion for a very slow rotaposi-tion velocity and for a given applied torque. Its characteristics depend on the instantaneous sit-uations of the meshing tooth pairs. Under load at very low speed (static transmission error), these situations result from tooth deflections, tooth surface modifications, and manu-facturing errors. Under operating conditions, STE generates dynamic mesh force transmitted to shafts, bearings, and to the crankcase. The vibratory state of the crankcase is the main source of the radiated noise [2]. To reduce the radiated noise, the peak-to-peak amplitude of the STE fluctuation needs to be minimized by the mean of tooth modifications. It consists in micro-geometrical modifications listed below and displayed onFigure 1:

(i) tip relief magnitudexrel,i, that is, the amount of

ma-terial removed on the tooth tip,

(ii) start relief diameter Φrel,i, that is, the diameter at

which the material starts to be removed until the tooth tip. Linear or parabolic corrections can be done,

(iii) added up crowning centered on the active tooth widthCβ,i/ j.

Many authors [3–11] worked on the optimization of tooth modifications in simple mesh systems. Only few of them [12–14] considered multimesh systems as cascade of gears where idler gear modifications aﬀect two meshes.

In this paper, the application is done on a cascade of three helical gears, displayed onFigure 2, for a total of 8 parameters (tip relief and start diameter for the relief for each gear, and added up crowning for a pair of meshing gears). Multipa-rameter optimization can easily become a diﬃcult task if the algorithm used is not well adapted. We will show that the Particle Swarm Optimization (PSO) fits eﬃciently with that kind of problematic. Indeed, it permits to select a set of solutions more or less satisfying in the studied torque

(3)

2 Advances in Acoustics and Vibration

b

Cβ,i φrel,i

Xrel,i

Figure 1: CrowningCβ,i/ j, tip reliefxrel,i, and start relief diameterΦrel,i.

range. Moreover, the robustness of the optimized solutions is studied regarding large manufacturing errors, lead, and involute alignment deviations. An additional diﬃculty arises because the modifications performed have to be eﬃcient on a large torque range. The dispersion associated is the source of the strong variability of the dynamic behavior and of the noise radiated from geared systems (sometimes up to 10 dB [15,16]).

2. Calculation of Static Transmission Error

The calculation of STE is relatively classical [17]. For each positionθ of the driving gear, a kinematical analysis of the

mesh allows determination of the theoretical contact line on the mating surfaces of gearing teeth within the plane of action.

Equation system which describes the elastostatic defor-mations of the teeth can be written as follows [17]:

Hu,F_(ω₌₀₎_·_F₌_δ(θ)₋_e₋_hertz(F),

Fi=F.

(1)

The following data are needed to perform this interpola-tion:

(i) initial gaps e between the teeth: they are function of the geometry defects and the tooth modifications, (ii) compliance matrix Hu,F_{, of the teeth coming from}

in-terpolation functions calculated by a Finite Element model of elastostatic deformations,

(iii) Hertz deformations hertz, calculated according to Hertz theory.

The calculation of the actual approach of distant teethδ on the contact line for each positionθ permits to access the

time variation of STE and its peak-to-peak amplitude Epp,

as a function of the applied torque (or the transmitted load

F) and the teeth modifications. We chose linear correction

Figure 2: Cascade of the 3 helical gears studied: 50 teeth/72 teeth/54 teeth.

for tip reliefs and parabolic correction for the crownings. All the modifications allow to reduce the STE fluctuation. The most influent parameter is the tip relief magnitude. Indeed, removing an amount of material on the tooth tip permits to make up for the advance or late position of the tooth induced by elastic deformations.

For the robustness study, the manufacturing errors are also considered and displayed onFigure 3. The manufactur-ing is not directly parameters of the optimization but as they have an eﬀect on the STE fluctuation they must be considered in the robustness study.

(i) Lead deviation: fHβ,i/ j = fHβ,i+ fHβ, j,

(ii) Involute alignment deviation: fgα,iand fgα, j.

A fitness function f to minimize is defined as the

(4)

Theoretical profile Actual profile

fgα< 0

b

fHβ,i

Figure 3: Involute alignment deviationfgαand lead deviationfHβ.

[Cmin−Cmax] approximated by Gaussian quadrature with 3

points. fi, j = Cmax Cmin Epp(C)dC−→ i=3 i=1 aiEpp(Ci). (2)

The fitness function of the whole cascade is then

f = fi, j+ fk, j. (3)

We have thereby 8 parameters for the optimization leading to a combinatorial explosion. Meta-heuristic methods allow an eﬃcient optimization, and we chose the Particle Swarm Optimization [18]. Obviously in that kind of problematic, the aim cannot be to access to the optimum optimorum but only diﬀerent local minima whose performances can be quickly estimated over the torque range by a home-built gain function G0=10 log10 fSi fref , (4)

where frefcorresponds to the value of the fitness function for

a standard nonoptimized gear.

3. Particle Swarm Algorithm

The principle of this method is based on the stigmergic behavior of a population, being in constant communication and exchanging information about their location in a given space [18]. Typically bees, ants, or termites are animals functioning that way. In our general case, we just consider particles which are located in an initial and random position in a hyperspace built according to the diﬀerent optimization parameters. They will then change their position and their

V(t−1)

p(t−1)

pi

pg

V(t)

Figure 4: Particle Swarm algorithm representation.

speed to search for the “best location,” according to a defined criterion of optimization. It is commonly called the fitness function which has to be maximized or minimized depending on the problem.

For each iteration and each particle, a new speed and so a new position is reevaluated considering:

(i) the current particle velocityV (t−1), (ii) its best positionpi,

(iii) the best position of neighborspg.

The algorithm can thus be wrapped up to the system of (5) andFigure 4: V(t)=ϕ0V(t−1) +ϕ1A1 pi−p(t−1) +ϕ2A2 pg−p(t−1) , p(t)=p(t−1) + V(t−1). (5)

A1 and A2 represent a random vector of number between

0 and 1 and the parameters of these equations are taken following Trelea and Clerc [19–21]: ϕ0 = 0.729 and ϕ1 =

ϕ2=1.494.

4. Robustness Study

First the tolerance range D0of a solution x0has been defined,

using a vector Δx = {Δx1,Δx2,. . . ,ΔxN}, which takes in

account the parameters variability. The gears studied have a precision class 7 (ISO 1328). Moreover, the manufacturing errors distribution is considered to be uniform over the range, which is the worst possible case in. Lead and involute alignment deviations and torque variation are associated in a 14-dimensionnal vector as following:

Δx=ΔXdép,i,ΔΦdép,i,fgα,i,ΔCβ,i/ j,fHβ, i/ j,ΔXdép,j,ΔΦdép,j,

fgα, j,. . . ,ΔC_{β,l/ j},fHβ,l/ j,ΔXd´ep,l,ΔΦd´ep,l,fgα,l,ΔC

, (6) wherei, j, and l correspond to, respectively, the gears with

(5)

Table 1: Parameters ranges.

Number of teeth Z=54 Z=72 Z=50

Tip relief magnitude and tolerance [μm] [15–150]±15 [0–150]±15 [15–150]±15

Start relief diameter and tolerance [mm] [230–241]±0.46 [200–215]±0.46 [153–168]±0.40

Added up crowning and tolerance [μm] [8–40]±8 —

— [8–40]±8

Lead deviation and tolerance [μm] 0±32 —

— 0±32

Involution alignment dev. and tolerance [μm] 0±12 0±12 0±12

0 10 20 30 40 50 0 100 200 300 400 500 600 STE (μ ra d ) Torque (N.m) Reference S1 S2 S3 S4 S5

Figure 5: Optimized and reference solutions versus applied torque - - - - torque range boundaries.

Then, the tolerance range D0can be written as

D0= x : x∈RN_|_x 0−Δx < x < x0+Δx . (7)

Contrary to the case studied by Sundaresan et al. [22], the robustness study concerns micro-geometrical modifications instead of macrogeometrical parameters (i.e., teeth number). The tolerance ranges are moreover noticeably larger than the ones considered by Bonori et al. [10], especially for the tip relief modifications. The fitness function cannot be assumed monotonic and the study of the extreme boundaries of the problem is not suﬃcient. The PSO is then used to locate the maximum of the fitness function in the hyper-space D0,in order to analyze robustness of the solutions. The

new values for the parameters which maximize the fitness function define the “degenerated solution,” noted xd:

xd∈ D0, f (xd)=max

f (x)|x∈D0

. (8) With this additional criterion, optimal solution corresponds to the less deteriorated rather than the minimalEpp.

5. Results

The cascade of three helical gears has to be optimized for torques from 100 Nm up to 500 Nm. A reference solution, with standard and not optimized tooth modifications, is used to emphasize the benefits of the Particle Swarm optimization.

0 0.2 0.4 0.6 0.8 1 Reference S1 S2 S3 S4 S5 Xrel,54 ϕrel,54 Cβ54/72 Xrel,72 ϕrel,72 Cβ,72/50 Xrel,50 ϕrel,50

Figure 6: Optimized parameters of the solutions.

The PSO calculations have been performed using a pop-ulation of 25 particles and stopped when a precision of 10−2μrad for peak-to-peak amplitude Eppis reached. The

al-gorithm stops the calculation when no improvement is found 50 times successively. All the following results have converged after 250 to 400 iterations. That corresponds to 7500 to 10000 evaluations of the fitness function (instead of 1014

for a Monte-Carlo experiment).Table 1lists the parameters ranges.

In order to illustrate the optimization process, Figure 5

displays 5 selected solutions—S1 to S5—corresponding to 5 local minima among the computed ones which all obviously are better than the reference solution in terms of minimal

Epp.Figure 6displays the optimized parameters of the

solu-tions rescaled in function of their extremum values.

According to the gain function (4), we can easily pick up the best solutions of the selected ones. Following the results listed inTable 2, solution S5, which provides−4.2 dB of im-provement compared to the reference solution, should be se-lected.

(6)

Table 2: Gain of the computed optimal solutions compared to the reference solution. Configuration Gain G0[dB] S1 −1.6 S2 −1.9 S3 −3.3 S4 −3.7 S5 −4.2 S1 S2 S3 S4 0 50 100 150 200 250 0 100 200 300 400 500 600 STE (μ ra d ) Torque (N.m) Reference S5

Figure 7: Degenerated solutions versus applied torque - - - - Torque range boundaries.

Figure 7displays the deteriorated solutions.

The first analysis of the deteriorating capacity of the solutions can be done using gain function (9) and listing results inTable 3: G1=10 log10 fnon−deteriorated fdeteriorated . (9)

The deteriorated reference solution has a gain of +6.7 dB compared with the initial reference solution. The solution S5 is worse considering the gain function (9), but its fitness function value is still less than the deteriorated reference so-lution one. On the other hand, the previous selected soso-lution S4 appears as the best one with only +2.3 dB of deterioration in the gain function (9) sense.

The second analysis of the deteriorating capacity of the solutions can be done using gain function (10) and listing results inTable 4: G2=10 log₁₀ fSi, deteriorated fref, deteriorated . (10)

The solution S1 emphasizes the importance of consider-ing the deterioratconsider-ing capacity. Indeed, although the optimal solution brings an improvement compared to the initial reference solution, it is likely to be less eﬃcient taking in account the possible manufacturing errors. The previous choice has to be reconsidered. On the other hand, the solu-tion S4 provides a good improvement of−3.7 dB compared to the reference solution and is quite robust as a gain of

−6.2 dB is observed if S4 deteriorated solution is compared with the deteriorated reference solution.

Table 3: Gain of the degenerated solutions compared to optimal solutions. Configuration Gain G1[dB] Reference +6.7 S1 +11.3 S2 +6.0 S3 +6.1 S4 +2.3 S5 +11.3

Table 4: Gain of the degenerated solutions compared to the refer-ence degenerated solution.

Configuration Gain G2[dB] S1 +2.8 S2 −2.6 S3 −4.2 S4 −6.2 S5 −0.4

6. Conclusion

Optimization with an eﬃcient heuristic method (Particle Swarm) has been done to determinate optimized parameters of a multimesh problem. The algorithm permits the gath-ering of many solutions which all lead to really satisfying results over the torque range studied thank to an integration of STE peak-to-peak amplitude by Gaussian quadrature. Finally, a robustness criterion has been defined based on the deteriorating capacity of the solutions which permits to do a more accurate choice about the optimal tooth modifications. Indeed, there are many ways of estimating the robustness of the solutions. In some industrial point of view, a solution which is less eﬃcient than another but much more robust should be preferably chosen.

Acknowledgments

This work has been supported by ANR (National Research Agency, contract number: ANR-08-VTT-007-02), ADEME (French Environment and Energy Management Agency), and Lyon Urban Trucks&Bus competitiveness cluster. The authors acknowledge gratefully this support and especially thank Denis BARDAY from Renault Trucks Company for his inestimable help.

References

[1] L.S. Harris, “Dynamic loads on the teeth of spur gears,” in Proceedings of the Institution of Mechanical Engineers, vol. 172, pp. 87–112, 1958.

[2] D. B. Welbourn, “Fundamental knowledge of gear noise— a survey,” in Proceedings of the Conference on Noise and Vibrations of Engines and Transmissions, vol. C177/79, pp. 9– 29, Cranfield Institute of Technology, July 1979.

(7)

[3] M. S. Tavakoli and D. R. Houser, “Optimum profile modifica-tions for the minimization of static transmission errors of spur gears,” Journal of Mechanism, Transmissions, and Automation in Design, vol. 108, no. 1, pp. 86–95, 1986.

[4] A. Kahraman and G. W. Blankenship, “Eﬀect of involute tip relief on dynamic response of spur gear pairs,” Journal of Mechanical Design, vol. 121, no. 2, pp. 313–315, 1999. [5] K. Umezawa, H. Houjoh, S. Matsumura, and S. Wang,

“Investigation of the dynamic behavior of a helical gear system dynamics of gear pairs with bias modification,” in Proceedings of the 4th World Congress and Gearing and Power Transmissions, vol. 3, pp. 1981–1990, Paris, France, 1999. [6] M. Beghini et al., “A method to define profile modification of

spur gear and minimize the transmission error,” in Proceedings of the Fall Technical Meeting of the American Gear Manufactur-ers Association (AGMA ’04), Milwaukee, Wis, USA, October 2004.

[7] M. Umeyama, “Eﬀects of deviation of tooth surface errors of a helical gear pair on the transmission error,” Transactions of the Japan Society of Mechanical Engineers, Part C, vol. 61, no. 587, pp. 3101–3107, 1995.

[8] R. Guilbault, C. Gosselin, and L. Cloutier, “Helical gears, eﬀects of tooth deviations and tooth modifications on load sharing and fillet stresses,” Journal of Mechanical Design, vol. 128, no. 2, pp. 444–456, 2006.

[9] A. Kahraman, P. Bajpai, and N. E. Anderson, “Influence of tooth profile deviations on helical gear wear,” Journal of Mechanical Design, vol. 127, no. 4, pp. 656–663, 2005. [10] G. Bonori, M. Barbieri, and F. Pellicano, “Optimum profile

modifications of spur gears by means of genetic algorithms,” Journal of Sound and Vibration, vol. 313, no. 3–5, pp. 603–616, 2008.

[11] S. Kurokawa, Y. Ariura, and M. Ohtahara, “Transmission errors of cylindrical gears under load—influence of tooth profile modification and tooth deflection,” in Proceedings of the 7th International Power Transmission and Gearing Conference, DE 88, pp. 213–217, American Society of Mechanical Engi-neers, Design Engineering Division, San Diego, Calif, USA, October 1996.

[12] T. Erltenel and R. G. Parker, “A static and dynamic model for three-dimensional, multi-mesh gear systems,” in Proceed-ings of the ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (DETC ’05), vol. 5, pp. 945–956, Long Beach, Calif, USA, September 2005.

[13] H. Vinayak, R. Singh, and C. Padmanabhan, “Linear dynamic analysis of multi-mesh transmissions containing external, rigid gears,” Journal of Sound and Vibration, vol. 185, no. 1, pp. 1–32, 1995.

[14] J. Lin and R. G. Parker, “Mesh stiﬀness variation instabilities in two-stage gear systems,” Journal of Vibration and Acoustics, vol. 124, no. 1, pp. 68–76, 2002.

[15] N. Driot, E. Rigaud, J. Sabot, and J. Perret-Liaudet, “Allocation of gear tolerances to minimize gearbox noise variability,” Acta Acustica United with Acustica, vol. 87, no. 1, pp. 67–76, 2001. [16] N. Driot and J. Perret-Liaudet, “Variability of modal behavior

in terms of critical speeds of a gear pair due to manufacturing errors and shaft misalignments,” Journal of Sound and Vibra-tion, vol. 292, no. 3–5, pp. 824–843, 2006.

[17] E. Rigaud and D. Barday, “Modelling and analysis of static transmission error of gears: effect of wheel body deformation and interactions between adjacent loaded teeth,” Mécanique Industrielle et Matériaux, vol. 51, no. 2, pp. 58–60, 1998.

[18] R. C. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in Proceedings of the 6th International Symposium on Micro Machine and Human Science, pp. 39–43, IEEE Service Center, Nagoya, Japan, October 1995.

[19] I. C. Trelea, “The particle swarm optimization algorithm: convergence analysis and parameter selection,” Information Processing Letters, vol. 85, no. 6, pp. 317–325, 2003.

[20] M. Clerc, “The swarm and the queen: towards a deterministic and adaptive particle swarm optimization,” in Proceedings of the International Conference on Evolutionary Computation (ICEC ’99), pp. 1951–1957, IEEE, Washington, DC, USA, 1999.

[21] M. Clerc and J. Kennedy, “The particle swarm—explosion, stability, and convergence in a multidimensional complex space,” IEEE Transactions on Evolutionary Computation, vol. 6, no. 1, pp. 58–73, 2002.

[22] S. Sundaresan, K. Ishii, and D. R. Houser, “A robust opti-mization procedure with variations on design variables and constraints,” Engineering Optimization, vol. 24, no. 2, pp. 110– 118, 1995.

(8)

Submit your manuscripts at

http://www.hindawi.com

VLSI Design

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Machinery

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation

http://www.hindawi.com

Journal of

Engineering

Volume 2014

http://www.hindawi.com Volume 2014

Shock and Vibration

http://www.hindawi.com Volume 2014 Mechanical

Engineering Advances in

Civil Engineering

Advances in

Acoustics and VibrationAdvances in

Electrical and Computer Engineering

Journal of

http://www.hindawi.com Volume 2014 Distributed Sensor Networks International Journal of

The Scientific

World Journal

Sensors

Journal of

Modelling & Simulation in Engineering

http://www.hindawi.com Volume 2014 Active and Passive Electronic Components

http://www.hindawi.com Volume 2014 Chemical Engineering International Journal of Control Science and Engineering Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014 Antennas and

Propagation

International Journal of

http://www.hindawi.com Volume 2014 Navigation and Observation International Journal of Advances in OptoElectronics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Robotics

Journal of